Tools / Physics · Astrodynamics
100% client-side · Kepler's third law · two-body circular modelHow long does one orbit take? Compute the orbital period from the semi-major axis or the altitude above a planet — or invert the formula to find the orbit radius for a target period, e.g. geostationary orbit.
For low orbits: a = R + h, then Kepler's third law. Works for any central body — pick the Moon or Mars and the right μ and R are used automatically.
Inverting Kepler: a = (μ(T/2π)²)¹ᐟ³. Use it to find where a satellite must fly to repeat its ground track with a given period — e.g. geostationary orbit.
| Body | μ (km³/s²) | R (km) |
|---|---|---|
| Sun | 132,712,440,018 | 695,700 |
| Earth | 398,600.4418 | 6,371 (mean) |
| Moon | 4,902.8000 | 1,737.4 |
| Mars | 42,828.375214 | 3,389.5 |
| Jupiter | 126,686,534 | 69,911 |
μ values are the standard gravitational parameters (GM) consistent with our Hohmann Transfer tool; 1 AU = 149,597,870.7 km (IAU 2012, exact); years shown are Julian years of 365.25 days.
Kepler's third law: T = 2π√(a³/μ), where a is the semi-major axis (orbit radius for circular orbits) and μ = GM is the central body's standard gravitational parameter. Example: the ISS flies at h = 408 km, so a = 6371 + 408 = 6779 km; with Earth's μ = 398600.4418 km³/s² the period is T = 5554.68 s = 92.58 minutes — about 92.6 minutes, or 15.55 orbits per day.
μ = G·M is the product of the gravitational constant and the central body's mass — the single number that sets the timescale of every orbit around it. This calculator uses: Sun 1.32712440018×10¹¹, Earth 398600.4418, Moon 4902.8000, Mars 42828.375214, Jupiter 126686534 km³/s², with mean radii 695700 / 6371.0 / 1737.4 / 3389.5 / 69911.0 km. The μ values match our Hohmann Transfer tool so results are consistent across both pages.
Use Mode 2: the orbit radius is a = R + h (planet radius plus altitude), then Kepler's third law applies. Example: a satellite 300 km above Mars (R = 3389.5 km, a = 3689.5 km) has T = 6804.02 s = 113.40 min. For comparison, 1000 km above Jupiter gives 175.68 min and 100 km above the Moon gives 117.79 min — the Moon's weak gravity makes even a low orbit take nearly two hours.
In Mode 2, altitudes below 160 km around Earth trigger a drag warning: atmospheric drag there decays an orbit in days to weeks, so a real satellite needs regular reboost or a higher orbit. The 160 km line is a rule of thumb — the Kármán line at 100 km marks where aeronautics ends, but measurable drag reaches well above it. (The warning applies to Earth only; other bodies have different or no atmospheres.)
Use Mode 3 (inverse): a = (μ(T/2π)²)¹⁄³. The GEO preset uses one sidereal day, T = 86164.0905 s, giving a = 42164.17 km around Earth — an altitude of 35793.17 km above the mean radius (35786.03 km above the equator). Note: matching the period is necessary but not sufficient for a truly geostationary orbit — it must also be equatorial and prograde; otherwise it is merely geosynchronous.
Scope boundary: this page's focus is the period T. The velocity v = 2πa/T (e.g. 7.66807 km/s for the ISS) is shown only as a secondary number for convenience. A dedicated Orbital Velocity calculator covers v = √(μ/r) as its main focus — same physics, different question. Keeping one focus per tool keeps each page the best answer for its keyword.
Because the calculator is an honest two-body approximation. At a = 384400 km with Earth's μ alone it gives 27.4519 days; adding the Moon's own μ (the proper two-body treatment) gives 27.2846 days. The real sidereal month varies around 27.32 days because the orbit is elliptical and the Sun perturbs it — effects outside this calculator's circular two-body model, stated openly in the limits box.
Different quantity, related by exactly √2: escape velocity v_e = √(2μ/r) is the speed needed to leave forever, while circular orbital velocity is v = √(μ/r) — so v_e = √2 × v_orb at the same radius. Use the Escape Velocity tool (#1613) when the question is 'how fast to escape', this tool when the question is 'how long is one orbit'.
The Hohmann Transfer tool (#1658) answers 'how do I move between two orbits' — it computes the two Δv burns and the transfer time of the elliptical path. This tool answers 'how long is one lap of a single orbit' — the period T. Different question, different formula, different page.
Input: semi-major axis in km or AU (1 AU = 149597870.7 km, exact), altitude in km, target period in s, min, h, days or years. Output: the period in all five time units at once (seconds, minutes, hours, days, Julian years of 365.25 days), plus the orbit radius, altitude and the derived velocity in km/s and m/s.
Circular orbits around a point mass in a pure two-body system: no atmospheric drag (except the explicit sub-160 km Earth warning), no J₂ oblateness, no third-body or radiation-pressure perturbations. Elliptical orbits need the vis-viva equation and are not covered. Treat results as design reference, not a substitute for a real propagator.
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